The function \(F_E\)  is the standard affine linear map between the reference element and \(E\) . The function \(F_e\)  maps from the reference element onto the smooth element \(e\subset \Omega \)  and is \(\mathcal {C}^1\) . For details, see Elliot&Ranner (2013), [1]}. On the surface both definitions coincide.
